NO9500006 UNIVERSITY OF OSLO DEPARTMENT OF PHYSIGS
Lower Hybrid Wave Cavities Detected by the FREJA-satellite
H.L. Pécseli,1 K. Iranpour," 0. Holter,1 J. Truisen,2 B. Lybekk,1 J. Holtet,' A. Eriksson,13 and B. Holback3
Department of Physics University of Oslo, Box 1048 N-0316 Oslo 3, Norway
UIO/PHYS/94-2S Received: 1994-12-12 ISSN-0332-S571 REPORT SERIES
i
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^fcfetof^s*:, Lower Hybrid Ware Cavities Detected by the FREJA-satellite
H.L. Pécseli,1 K. Iranpour,1 0. Holter,1 J. Trulsen,2 B. Lybekk,' J. Holtet,' A. Eriksson,13 and B. Holback3
Department of Physics University of Oslo, Box 1048 N-0316 Oslo 3, Norway
MO/PHYS/94-25 Received: 1994-12-12
ISSN-0332-5571 Q u (i £> L. ; £~
1 Plasma and Space Physics Group, Department of Physics, University of Oslo, Box 1048, Blindern, N-0316, Oslo, Norway
2 Department of Theoretical Astrophysics, University of Oslo, Box 1079 Blindern, N-0315, Oslo, Norway
•' Swedish Institute of Space Physics, Uppsala Division, S-755 91 Uppsala, Sweden Lower hybrid wave cavities detected by the FREJA-satellite
H. L. Pécseli1, K. Iranpour1, 0. Holter1, J. Trulsen2,
B. Lybekk1, J. Holtet1, A. Eriksson3 and B. Holback3
1 University of Oslo, Physics Department, Plasma and Space Physics Box 1048 Blindern, N-0316 Oslo, Norway 2 University of Oslo, Institute for Theoretical Astrophysics Box 1029 Blindern, N-0315 Oslo, Norway 3 Swedish Institute of Space Physics, Uppsala Div. S-755 91 Uppsala, Sweden
Abstract
Localized electrostatic wave packets in the frequency region of lower-hybrid waves have been detected by the instruments on the FREJA satellite. These waves are usually associated with local density depletions indicating that the structures can be interpreted as wave filled cavities. The basic features of the observations are discussed. Based "n simple statistical arguments it is attempted to present some characteristics which 'iave to be accommodated within an ultimate theory describing the observed wave phenomena. An explanation in terms of collapse of nonlinear lower-hybrid waves is discussed in particular. It is argued that such a model seems inapplicable, at least in its simplest form, by providing time and length scales which are not in agreement with observations. Alternatives to this model are presented.
1 1 Introduction
1.1 The wave experiment on the Freja satellite
The Freja satellite was launched into an elliptical orbit with an apogee of 1750 km and a 600 km perigee. An inclination of the orbit plane of 63° will give crossings of the auroral oval at slant angles or at tangential intersections. The spacecraft will thus stay inside auroral forms for long periods of time, and scan extended longitude/local time ranges. The main aim of the project was to study microphysics of the auroral acceleration regions. For an overview of the project see Lundin et al. (1994).
The Freja wave experiment (F 4) measures AC variations in the electric and magnetic field, electron density and temperature, in addition to monitoring the background plasma density and electron temperature. The spacecraft has an array of 6 wire booms with spherical probes in the spin plane. There are two long boom pairs, with probe separation "-21 rn, and one shorter pair, with ~11 m probe separation. The probe separation of the long boom pairs corresponds to -~10G Debye lengths, and --10 and "-3 Larmor radii for protons and oxygen ions, respectively, for typical local plasma conditions. (The spacecraft will travel this distance in ~~3 ms.) The probes may either be operated in a voltage mode or in Langmuir mode, for electric field or density/temperature measurements, respectively. The /i/experiment may take its signal from one of the long boom pairs with a limited frequency range, • 1MHz, or from a dedicated A/-probe, spherical sensors with 1.2 m separation, at the full frequency range, 4 MHz Magnetic wave fields are measured by a three axis search coil experiment. The wave experiment also carries a dedicated cylindrical Langmuir probe.
All data are recorded as wave forms. To improve the time resolution the data are sampled at a rate much higher than the telemetry rate. Therefore, measurements are made in a snapshot modi-, where the data samples are stored intermediately in a buffer memory, and transmitted to the ground at the telemetry rate. The length of each data sample depends on the frequency bandwidth set by the experiment. The experiment has in principle four 'iffrrent frequency rnnges, kf with bandwidth 4 MHz (electric field only), mf: 16 kHz bandwidth (electric and magnetic field), // (four channels in parallel): 2 kHz bandwidth (electric and magnetic field, An and AT,) and Langmuir DC-mode with up to four signals in parallel with continuous sampling at 128 per second. The sampling duty cycles for the different bandwidths at normal telemetry rate are 0.01% for hf, 1.6% for mf and 18.8% for // The sampling times can be varied, and the maximum "snap shot times" are 1.024 ins for hf, 12") ms for mf and 24 s for //. To increase the probability of catching an event with lli*- /// !)()!?£ fluty cycle, many snap shots are taken within each cycle, and an automatic evaluation selects the "best" event, i e. the sample which shows the highest HMS paver The experiment can be operated in a great variety of modes, including burst modes, where a "i 'A Mbyte memory can be used to increase the sample lengths. For more details about the experiment and modes of operation, see Holback et al. (1994).
1.2 Observations of lower-hybrid waves
Localized electrostatic wave packets in the lower-hybrid wave frequency region have been detected by instrumented rockets MARIE as reported by LaBelle et al., (1986) and on TOPAZ III described by Vago et al. (1992), and lately by measurements from the FREJA satellite (Dovner et al. 1994, Eriksson et al 1994). Instruments onboard the FREJA satellite (Lundin et al. 1994) measure wave-bursts in the range of several hundred Hz to L. few kHz. A characteristic feature of these events is an accompanying density depletion as illustrated by the example in Fig. 1, giving the appearance of wave filled cavities in the ionospheric plasma. In the present work we summarize the basic properties of the observed phenomena and attempt to outline some characteristic features which must be accommodated within a theory which attempts to describe and explain the observed wave phenomena.
There seems to be no straight forward or simple explanation of the observations. Various scenarios can be considered; i) The lower-hybrid waves and the density depletions have diiferent and unrelated origins and the observed synchronization is a consequence of the variations of the wave-dispersion relation caused by the density variation, ii) The den sity variations are induced by nonlinear wave phenomena, where two basically different candidates can be proposed a) depletions caused by plasma expansion due to local wave particle interaction and b) a wave-collapse initiated either by a modulational instability or by nonlinearities of burst-like wave excitations. Other more complex phenomena can be envisaged, some being a combination of i) and ii).
The basic problem in the interpretation of the data is associated with the fact that only a number of individual phenomena are observed. Unlike in a laboratory experiment it is here not possible to make observations at various stages of the evolution to obtain the full dynamics of the problem. The observations from the spacecraft present individual events. The time evolution of the phenomena can however be compared to the ones predicted by a model by a statistical analysis, assuming that an event is intercepted at random and uniformly distributed times. First some basic statistical properties are derived from the model in terms of distributions which can then be tested against the available data. It is important that the phenomena discussed in the following occur sufficiently often to give basis for reliable statistical estimates.
It is interesting to note that spiky bursts of lower-hybrid waves are observed also in labo ratory experiments by for instance \ickenig and Piel (1987), and it is plausible that much can be learned about the ionospheric processes by a dedicated laboratory experiment.
Tliih report i* organized as follows: In sect 2 we discuss the basic characteristics of the
3 lower-hybrid waves, while sect. 3 contains model discussions for their interpretation. An interpretation in terms of collapsing lower-hybrid waves is one example, which is considered in detail. It is argued that model properties and observations are compared most easily in terms of the statistical distributions of the cavity-widths. This analysis is presented in sect. 4. Section 5 contains discussions of the observations and the analytically obtained probability densities, in particular also comments on empty cavities (i.e. cavities without a corresponding wave component). Finally, sect. 6 contains our conclusions.
2 Basic wave characteristics
Simultaneous measurements of electric and magnetic field components (see Fig. 1) demon strate that the fluctuations considered here are electrostatic (Dovner et al. 1994, Eriksson et al. 1994). Only very small magnetic field fluctuations are observed and these show no correlation with the variations in electric field. The spectrum of the fluctuations have a well defined lower cut off frequency. Using the experimentally obtained values for plasma density and ambient magnetic field, it is readily demonstrated that the observed frequencies in the spacecraft frame of reference are comparable to the lower-hybrid frequency, CJLH- In a fluid description, the limit of small wavenumbers, the linear dispersion relation for these v..\.es is (see e.g. Shapiro et al. 1993)
i 1,, „,- IM (h\* ] WLH 1 + -(*/?)- + ; m 2(*r)2u£. I We used the standard notation, m (A/), Tt ('/',), a/p, (u-'^j and S2r, (fir,) for electron (ion) mass, temperature, plasma and cyclotron-frequencies, respectively. An approximation relevant for the present problem is u£ • • n*,. The expression {J) assumes one ion species only For many ion species plasma some mod ifications are necessary First of all, the lower hybrid frequency u//jf must be obtained as the larger solution to , . V v *"' i We assumed u>2 << Q£,. If in addition w2 >> H^j, the result is a _ Sj «y.j The latter approximation should be considered with care, though, when a light ion specie is present. In that case it might easily be that u^ ~ fl^y with j denoting the light ions. For multi-ion plasmas new solutions for u> between the ion-cyclotron frequencies will appear. These are sometimes also observed in the FREJA data, but they will not be discussed here. In the expression for the R2-coefficient the temperature and mass of the lighter ion specie should be substituted. Finally, the M/rn ratio in (1) should be replaced with UJ^J Ylj^j- The second term in the parenthesis of (1) accounts for the cold-plasma resonance cone. Eventually, as k —* 0, the lower hybrid waves transform to magnetosonic modes- The [ast term accounts for the dispersion in the long wavelength limit where the lower hybrid waves acquire an electromagnetic component. For the limit relevant here this term can be considered as a small correction to the cut-off in the purely electrostatic branch. Lower hybrid waves have strong directional dispersion with angle to the ambient magnetic field. Consequently an observation of a well defined frequency indicates a well defined direction of propagation. The wavenumber spectrum in that direction may however be relatively broad for the electrostatic waves, without giving rise to an appreciable broadening of the frequency spectrum. The relation (I) can be approximated in terms of the actual plasma parameters as 2 2 7 2 / . .— = 3853 (l -f 0.049k + I348(%'*) - 2.8 • 1(T /* ) t (2} in MKS-units, where we used a mixture of //+ and 0+ in the ratio 3:1, and where B = 0.4-10 4 Tfi.e. 0.4 Gauss), n = 5 10s m'3 and the bulk plasma temperatures F, =- T, - 0.2 eV. The J value For the plasma is I0"8 for the present conditions, but may vary by a factor of 10 '2 102 with other relevant plasma densities and temperatures. Losses are not included as the plasma in cnllisiouless to a very good accuracy. For the present conditions, we find the collisional mean-free-path Cw - 64irnAø/ ln(8:rnA£,) ~~ 2 • I0a m for deflections at an angle larger than 90°. In Fig. 2 we show the frequency, U//2TT, as a function of wavenurnbcr. This result is obtained by direct numerical solution of the full kinetic plasma dispersion relation. The result agrees well with (2) for small wavenumbers. In particular the kinetic damping is negligible The electromagnetic correction, corresponding to the last term m (2), is responsible fur the bend-over in the dispersion surface as A: • 0 For wavenumbers of interest in the present study this correction is vanishingly small Significant deviations appear when the frequency approaches one of the ion cyclotron harmonics for large fcx and wave dispersion is influenced by the ion Bernstein modes. For the situation depicted in Fig 2. witli a proton gyro frequency of 610 Hz, this is not a problem We have 5 investigated the full dispersion-relation in detail, but this analysis is outside the scope of the present paper. Unfortunately, the plasma parameters are not known with sufficient accuracy to determine the actual wavenumbers from measured frequencies. Consequently, the phase velocity of the waves cannot be determined accurately because of the uncertainty of the Doppler shift, and the frequency in the rest frame is therefore uncertain. It is however possible to estimate the wavenumber k which gives a Doppler shift k- V æ U/LH\ with a typical satellite velocity V == 7 • 103 m/s we find k == 1.4 m"1 corresponding to a wavelength A ^ 4.5 m which is close to a typical ion Larmor radius. The actual lower hybrid wavelengths are probably longer than this. As a consequence the measured frequency u> — u»' -f k • V in the satellite frame will get its dominant contribution from the actual frequency u>' in the rest frame, rather than the Doppler shift. It is possible only in some special cases to give an estimate of actual wavelengths of the lower-hybrid waves. Another consequence of the uncertainty on the wavenumber is that the measured wave amplitude is not easily related to the actual one. The measured value is determined from the potential difference between two probes, 21 m apart. Ideally, for wavelengths much longer than this value, the measurement gives the electric field component in the direction determined by the probe separation. In reality, the measurement probably under-estimates the actual electric field as we expect relevant wavelengths not to be longer than a few times the probe separation. The generating mechanism for the lower-hybrid waves will not be discussed in detail here, bul we note that in many cases it can be argued that they are generated by linearly unstable anisotropic (for instance loss-cone or ring-shaped in velocity space) ion velocity distribution functions (Rosenbluth and Post, 1965). Alternatively, lower hybrid waves can also be excited by tliin cross field current sheaths having a width smaller than the ion Larmor radius- In that case the electrons will E • B-drift across magnetic field lines while the inn component remains essentially immobile. The resulting currents excite waves in the lower hybrid frequency regime (McBride et al. 1972), propagating within a narrow angle essentially perpendicular to B with k^/k == (m/A/)1'2. The currents themselves can be difficult to detect directly by the spacecraft since the probability of actually crossing them is small, and the magnetic field generated by them may not be easy to detect. Lower hybrid waves can be induced also by parametric decay of other waves but such processes can account, at most, for a minority of the observations due to a lacking candidate in the data for large amplitude driving wavemodes. We have in mind that the instabilities and decay processe.s often give rise to excitation of also other wavetypes, such as ion acoustic waves The data frequently give evidence for such waves, but often the lower hybrid waves are the only ones observed In these eases, the generation region might be far from the observation pom» *.. tint the..ther waves have been damped out before they could be observed. Linear ioTiver->i"ii ml" lower hybrid waves of other wavetypes at plasma inhotnogeneities appears iinlikiK I" < Mii-e "f the usually modest elliiiencies involved t\ Concerning the low-frequency density variations, the situation is somewhat different. Here signals are available from two Langmuir probes with an 11 m separation. In most of the cases, a density depletion is observed in both signals, and it is found that the depletion propagates from one probe to the other, generally with only modest change in shape. The absolute velocity of cavities can be determined (Dovner et al. 1994), indicating that they are on average almost stationary in the rest frame, although the uncertainty of this estimate is comparable to the ion-sound speed. Drift of the plasma represent an uncertainty for this result. 2 A typical scale size of 50 m can be determined. Relative density variations of njnQ PS 5 • 10~ are common and can be up to 10"l. The average distance between cavities along the space craft trajectory is of the order of 150 m, but with a large statistical spread. The experimentally obtained probability density for the relative distance between cavities is shown in Fig. 3. Very large distances are under-represented in the data because of the finite sample duration (usually 0.75 s). A dotted line shows an exponential fit, exp(—x/£), where £ corresponds to approximately 150 m. This exponential fit indicates that the probability of finding a cavity in a small interval dx is proportional to dx itself, with a constant of proportionality given by the density, fi, of cavities along the spacecraft trajectory. In particular we should like to point out that a model with many randomly distributed, statistically independent, density depletions with a uniform spatial distribution, lesults in A a Poisson distribution, P{N) = (/i£) 'exp(-fi£)/iV!l for the number, TV", of cavities in a given interval, C, along the spacecraft trajectory. 2.1 Wavelet analysis In order to study the distribution of wave frequencies in a cavity in detail, we performed a wavelet analysis (e.g. Kaiser, 1994), with typical results shown in Fig. 4. In the present context we may consider the wavelet transform simply as a local Fourier transform performed in a systematic way to optimize the uncertainty relation for each frequency component. We note that the frequencies with large amplitude m the cavity are also present outside; we interpret this as evidence for propagating waves. The frequency distribution is relatively narrow, and best explained by a model assuming a wave-vector predominantly perpendicular to B with narrow distribution in directions. In a number of cases a frequency component is observed only instde the cavity, see Fig. 4b. These are likely to be trapped waves, and their wavelengths must then be comparable to or shorter than twice the cavity width. Thr gap in frequencies separating the trapped from the free modes in Fig. 4b might indicate that the trapped constituent corresponds to only one eigenfrequency of the cavity This and similar observations may help to estimate the local dispersion relation of the waves The trapped component usually has a smaller amplitude than the freely propagating one A possible implication of a trapped component is a dynamic process during the interaction between the lower-hybrid waves and the density depletions. It is nnhkrly that the trapped wave component is directly excited inside the cavity because of its comparatively smaller amplitude as compared to the propagating mode. A possibility for the excitation of the trapped mode is decay of the nropagating wave into an eigenmode of the cavity, i.e. decay from the continuum of modes to a discrete set. The dynamic process which gives rise to the trapping need not, however, be a non-linear interaction. It could in principle as well be a rapid depletion of local plasma density caused by mechanisms entirely independent of the wave component. Also such a process can give rise to trapped modes. 3 Model discussions 3.1 Ray-t rac ing In order to study the apparent "synchronization" of the lower hybrid waves and the density depletions we performed a ray-tracing simulation. The group velocity is obtained from (1), ignoring the last term, giving Vfco/(k) = vex tfk+^tM-A,,!) (3) The second term in (3) is perpendicular to k. A varying density, n — n(r), with B — const., implies a corresponding variation in &LH — ^tw(r). Considering the waves as propagating in a WKB-sense we argue that a lower hybrid wave packet will propagate in a time stationary plasma with a slow spatial variation so that the local dispersion relation (1) with uji,n - wttf(r) is satisfied with constant frequency. The density variation of R is immaterial in this context. This problem is analyzed most easily with a ray-tracing code where we solve the equations d,T = Vku,(k) and dtk - -VwLW(r). Results art* shown in Fig. 5 for three cases; a trapped, a marginally trapped and a free ray. Straight line segments branching off from the ray indicate the local wavevector by a direction and a length (in arbitrary units) at the given position. These markers are drawn at equidistant times, i.e. their relative distance is a measure of the local group velocity. Indications are that this velocity actually increases in a density depletion, but that the wave nevertheless spends more time there because of its "meandering". The result is an expected net increase in wave energy in the density depiction. If time variation of the density depletion is allowed, the analysis becomes somewhat more complicated where also trapped modes in the density depletion can be accounted for For the results in Fig. 5 it is evident that a density depletion has a pronounced effect on the propagation of lower hybrid waves These calculations assume the density depletion to be pre existing and do 8 not provide arguments for a feed back mechanism where the waves generate or influence the density depletion. The arguments in this section were based on a WKB-simulation for wavelengths much snorter than the cavity widths, but they will presumably remain valid also for wavelengths comparable to the scale size of the cavities. 3.2 Nonlinear models Evidently, the prime question is to estimate the wave energy density in order to assess the importance of nonlinear effects. In general the contributions from both the If- and m/-bands are important although for the example shown in Fig. 1 the energy is almost entirely in the 7n/-band. Unfortunately this poses a problem; because of its high sampling rate, the m/-band is transmitted in a burst mode of 15 ms duration, typically every two seconds. The likelihood of this burst to coincide with a cavity is small and only relatively few events are observed in this band. The //-band with its low sampling rate covers many, since here the burst duration is usually 0.75 s with 2 s intervals. On the basis of a modest number of optimum events where both band are available for determining the wave energy, an electric field amplitude in excess of 50 mV/m can be estimated. The aliasing effects mentioned before makes this a possible unife^-estimate for a plane wave. The wave energy density is obtained from the relation where K is the dielectric coefficient tensor. For the present purpose we obtain a sufficient accuracy by u,c -if the cold plasma approximation, where the components of K are given by e.g. Allis et al. 1963. The dielectric properties of the plasma contribute to the energy density with a factor depending on the polarization of the wave, i.e. the direction of the wave-vector with respect to the magnetic field. For the present plasma conditions, and with the wave vector essentially perpendicular to B, we find that this correction contributes with a factor O(l) A typical wave-energy density can be obtained as =s 2 • \0~14 J/m3. For 10 3 comparison, we have a typical thermal energy density no«Te ^ IO' J/m . To assess the importance of the non linearity we may also compare the pressure term 2 2 V(n«Tr) with the ponderomotive force (w^/u/r,//) ^ V £ acting on a plasma volume element. For n/n0 -^ 1-10% we find that the ponderomotive force can easily exceed ~- 1% of the thermal pressure; in extreme cases the two can be almost comparable. It therefore seems safe to conclude that nonlinear wave phenomena are indeed important for processes as those illustrated in Fig. 1, at least in a significant number of cases. This is our basic argument for preferring an interpretation of the data in terms of nonlinear wave processes. A wt df nonlinear model equations for the lower hybrid wave dynamics has been proposed 9 (Musher and Sturman, 1975, Sotnikov et al., 1978, and Shapiro et al. 1993). They are based on a set of nonlinear equations for the wave potential 2 _2i_L8^V- * rv+ -# (!fc) V -** = uJcfr m cz xWrt./ m * .^(^.ri).i (4) and (8] - C.!V2)^ = i f»"* V'(Vf • V*< • i. (5) The magnetic field is ignored in (5) for the slow plasma variation because it is anticipated that the appropriate time scale is shorter than the ion gyro-periods, which are approxi mately 16 ms for H+ and 26 ms for 0+ for the present conditions. The assumption has to be verified a posteriori. For time-stationary conditions this equation is reduced to -=iT~^ (V*- V*-)-». (6) which can be rewritten in terms of the phase shift, A, between the two transverse compo nents Ex and Ey of the electric field inside the cavity ~ = ~^ £,E„sinA. (7) Contrary to the case of Langmuir oscillations, the lower-hybrid waves can be localized with density wells, n < 0, as well as density "humps", n > 0, depending on the sign of sinA (Shapiro et al. 1993). The dominant (vector) nonlinearity retained in [A) originates from a high frequency density variation fi,% -~^~V(nVct> • B). This nonlinearity vanishes however for perturbations which are one dimensional in a plane perpendicular to the magnetic field, in which case the standard scalar nonlinearity has to be retained, even though it is of order U>LH/U>„ **„ 1 as compared to the vector non- linearity. Equation (4) can be considered as the full-wave equivalent of the ray tricing mode) discussed before. The shortcomings of the model equations are self evident, the equation (5) for the low frequency bulk plasma density entirely omits kinetic effects, for instance The assumption that the dynamics are characterized by a time scale much larger than tlie urn cyclotron frequency is sufficient in a fluid theory, but in a kinetic treatment harmonics of the gyro frequency become relevant also and the assumption ran n->t be eas ily generalized for these. For very low frequencies only the B parallel motion of the ions should be included on this time scale Also the kinetic effects for (he lower hybrid waves are ignored, the damping in particular. The model furthermore assumes a Maxwellian II) plasma, although this may be far from the case in many realistic conditions. Deviations from Maxwellian distributions can have important consequences, this is at least known to be the case for weakly nonlinear Langmuir waves (Treguier and Henry, 1972). In spite of these shortcomings, it is worthwhile to use the results from these model equations as a guide-line for which nonlinear wave phenomena to expect. The properties of the model equations (4)-(5) have been studied in great detail, see e.g. Shapiro et al. (1993). Most important is the observation of collapsing solutions where for sufficiently large wave intensities the nonlinearity forms a wave-filled plasma cavity, which subsequently collapses into a singularity within a finite time with a time variation of its diameter if LL^{tc~t) * and £„ ^ (tc - t) (8) where tc is the (arbitrarily chosen) collapse time. The cavity is strongly B-field aligned with L\\ > LkJM(m. There is no natural propagation velocity associated with the cavity. It is the purpose of the present study io investigate whether a nonlinear plasma wave phenomenon like 'he one outlined here can account for the observations. In the following, different aspects of the problem are discussed separately for convenience. In agreement with the analytical results based on (4) and (5) we would expect lower-hybrid wave fields to build up by some mechanism, e.g. a linear instability. When a sufficiently large amplitude has been reached, a modulational instability sets in, breaking the wavetrain up into wave cavities which collapse and ultimately dissipate the wave energy at small scales. In the case where the waves are generated at short wavelengths the model outlined here may have to be modified by inclusion of parametric decay processes. As mentioned before, it seems difficult to find support for interpretations in terms of parametric processes on the basis o[ the existing data. Two basically different scenarios can be envisaged; one where the lower-hybrid waves are continuously maintained, essentially rendering the caviton formation a statistically time stationary process. Alternatively we can imagine an event where lower-hybrid waves are excited in a large region of space as a burst, which eventually breaks up into cavitons and finally dissipates. It is not possible to discriminate between these two scenarios on the basis of the available data. However, large amplitude lower-hybrid waves are frequently observed in the //-band and it is bkely that they constitute a background wave component for extended periods of time. The set f»F equations {-\) and (5) predicts that lower hybrid waves trapped in cavities should radiate !<>w amplitude magnetosonic waves with long wavelengths. It is uncertain, however, to what extent this radiation is detectable. These waves will in general be of very long wavelength and a cavity may be a poor "radiator" for such conditions. The wave component transmitted out of the density depletion (the "escaping" wave) is expected io h iS'f a small amplitude Consequently, it can be possible to observe an apparent wave trapping which would be associated with an ideal electrostatic cut-off in a density cavity. 11 from Norwegian companies, may make the oil company lose some British "licensing points" which implies that the oil companies will have to balance carefully their decision of whether to buy from a Norwegian or a British supplier. 3. Britain as a "door-opener" to the rest of the English-speaking world One of the companies we interviewed emphasized the special importance of Britain as a "door-opener". If a Norwegian company has success in the British market, this will give the company a great deal of recognition. In other words, to succeed In British sector, may help making It easier to enter other foreign markets. In the Interview it was also stressed the fact that British culture is very dominant In much of the English-speaking world. Learning the British culture by working towards the British sector may thus prove to be helpful in foreign markets around the world. 4. The British sector is a market of considerable size For the Norwegian offshore supply industry the British sector represents a market of considerable size. The size combined with the near proximity to Norway should make British sector a very attractive market for Norwegian supply companies. However, it is also a market characterized by heavy competition from domestic suppliers. This combined with culture/communication problems and protectionism makes the British sector a tough market to enter. 3.2 Barriers In the interviews most of the time was used to discuss the various problems that the companies had experienced in their attempts to enter the British sector. For companies that had not yet made any attempt we focused on what factors they believed they would encounter if they should try. Knowing that the customers, particularly, the oil-companies, are often the same in offshore-markets all over the world, we also made some Inquiries about experiences In other foreign offshore-markets than the British one. After completing the ten Interviews. It is our opinion that the barriers, or the sources of problems, can be divided Into three groups. We will present the Norwegian companies' views with regard to these constraining factors. 1. British culture/commiinlgaHnn problems Nearly all the company representatives claimed that culture/communication problems had made It difficult to enter the British sector successfully. Some of the dlflicultles may stem from language problems. Although most Norwegians speak, read and write English pretty well. It Is always difficult to do business In a foreign language, especially if the counterpart Is very 9 As the satellite moves (essentially in the direction perpendicular to B) we expect predom inantly large scales to be detected, observations of the actual collapse being statistically improbable. Although the collapse itself is thus unlikely to be detected (its "cross-section" is too small), the time evolution preceding it will in principle be evidenced by the proba bility distribution of length scales. It is important here to take into account also the fact that the probes on the space craft can intersect a given cavity at different positions. The observation gives a "chord-length" not to be confused with the diameter, i.e. the obser vations would give a statistical distribution of these lengths even identical and stationary cavities were randomly distributed in space. This should of course be taken into account also in other contexts where density cavities are studied (Bostrom et ah, 1989). 4.1 Cylindrical model As an illustration assume first that the cavity has a B-elongated cylindrical shape with a circular cross section of radius ^LL{t) which is time-varying during the collapse, while L\\ is for the moment considered infinite. The observations are then essentially restricted to a plane. The geometrical cross-section for the cavity is thus a — L±_ with dimension "length" in this planar approximation. The probability for actually encountering a cavity in an interval of length dx along the spacecraft trajectory is proportional to p(L± )L± dL^ dx, where, again, ^(Li )dL L is the density of cavities having the diameter L within a small length interval around the actual value for L^. Assuming the cavitie^ to appear and disappear at random, the dynamics can be assumed to be time-stationary in a statistical sense, and fj(LL) is constant even if Li varies with time f>r the individual cavity. In a given realization there are many different scale sizes present at the same time. The relative probability for encountering one particular value for LL is given by its relative density. Assuming that the spacecraft has encountered a cavity it is evident that the probability of its diameter being L± is proportional to the density of cavities with that particular diameter as well as to the corresponding geometrical cross-section. With the expressions derived before we have the probability density for the diameters for observed cavities P{Li) - S^i/^lmax ^or ^J- < ^imax and zero otherwise. \M the angle between the spacecraft trajectory and the magnetic field be &. A straight-line trajectory intercepts cavities of length scale Lj. along cords with y ~ const, in the ellipse *a . v2 __ i LI/sin2 0 L\ 4' with all y- values in the interval { - \L^, \LL) being equally probable. Allowing for 6 4- 90° the model is sufficiently general, even with the assumption that the B-perpendicular cross section of the cavity is circular. The distribution of cord lengths. f, is readily obtained as tsw e mi^=~t ' , do 13 for 0 < t < L±/sm$ for a given fixed LL. The angle 0 is considered as a constant. It was here assumed that the satellite speed is so large that the cavity does not change appreciably during its passage. Evidently (10) predicts that there is a large probability of finding t ~ L±/sm0. For distributed cavity diameters Lj_ we have P(l) = r P(e\L )V(L )dL Jo ± ± ± 2 V = tsm 9 . -.dL± (11) 2 2 2 Je,in9 L±y/L ±-£ sm 0 with the actual probabihty density V(L±) for the cavity widths, Lj_, to be inserted. The final result, within this model, for the probability density of observed cord lengths becomes PW = ^ (12) Hi xmax/ for isinB < L±max- This result is shown in Fig. 7a. Evidently, the limit 9 ^ 0° is inapplicable in the present B-field aligned model. In case there are reasons to expect that cavitons start out with a significant distribution in initial scale sizes, i-imaxi the averaging over the appropriate probability density is easily included in (12). 4.2 Ellipsoidal model The foregoing result for P(i), the probabihty density of observed cord lengths, was derived for a cylindi ical form of the cavity. Somewhat more generally we can assume a cavity of the form of a rotationally symmetric ellipsoid with major and minor axes L^ and L±, respec tively. Because of the rotational symmetry of the problem with respect to the magnetic field this is probably a quite adequate model. The cross-section of the cavity for a space craft moving at an angle & to the major axis (which is parallel to B) is L\ + Lf. tan *(£.., £ ) = KL x L 1 + tan2 $ The probability of encountering a certain cavity specified by (£|j, LL), is proportional to Ø{L\\,LL) and to the appropriate density of cavities. In determining F{Li,L\\) one can not offhand equate it to P{L\_ )P(L\\) since the time evolutions of L{i and tj. are not statistically independent. Referring to (8) we take, in a given realization, L\\ - (3L\ where the constant 0 is determined by the initial conditions. Using (9) as well as the cross section obtained previously, we obtain L\ +• ^tan'e 14 L L 2 fL±majc±ma.x r-r amax h\ + Ljj tan 9 6m-pLl)L' 2 dL±Li\ Jo Jo L 1 + tan 6 where jCymax — Æ^lmax- The normalizing quantity is solved as 1 2 2 2 2 3/2 [2 + (3 0 1 imax tan 9 - 2) (l+ /? £imax tan *) | x fl5/34 tan4 Oy/[l + tan2 0)1 . Now we consider the probability density for observed cord-lengths for a given cavity spec ified by Z/||, L±. After some calculations we obtain P(t |£j.,i.|) = 2/ (13) h\l\ 2 2 2 for 0 < I < LHLX/C with C = Li cos 0 + Ljjsin S. For B = 90° we have P(t | £j.,I(|) = 2£/L\ as a particular case of (13) independent of L\\. For distributed values of Lx. and £|| we find />(/) just as in (11) 2 COS sin fl\ P(l) = 2f || V(Li.,Lv)dl1.dLlv (14) 2 2 2 with the integration restricted to that pr.rt of the Lj.,Z.j|-plane where IÅL\ > t {L L cos 6+ z ani £j|sin 0) and £1 < £j_max> £[| < £||max- The initial values of L\\ and LLi i.e. i||max ^ timaxi afe here assumed given. In the actual case they will differ from case to case, and the corresponding probability density must be included in the analysis at a later stage. We consider a limiting case, which is relevant for the data, where 9 ~ 90°, i.e. the satellite propagates essentially perpendicular to B. In this limit the expressions are considerably simplified. Inserting the previously obtained P{LLl L\\) we obtain Pin = 4 (15) IA V^imax / This result is shown in Fig, 7b with a full Une. This result af well as that in Fig. 7a is appli cable for the case where the spread in the initial values of £i.max is small. The dashed line on the figure indicates the corresponding result for a targe spread, where we, for the sake of : eJc argument, assumed a probability density of the form 8/(£3>/fl )(£,J.max/£)* p(~(^imax/0 where C is a typical length scale for the initial conditions. Now, (12) as well as (15) assigns a finite probability to measuring a very small cavity width, corresponding to for instance cases when the satellite trajectory crosses almost at the boundary of the cavity. In reality such cases are unlikely to be properly recognized in the background noise level and these contribi! ' >ns will be underrcprescnted in the experimental estimate for the probability density. 15 4.3 Comparison with observations It is interesting that (12) as well as (15) gives a flat maximum for P{t) for t in the range ~- (\ — l)£±max> see Fig. 7. The distribution obtained on the basis of data is shown in Fig. 8, where a chord length is identified from the separation between the two points of maximum curvature at the baseline of the signal. The results in Fig. 8 are not supporting an interpretation in terms of (12) or (15). Even with reservations due to statistical uncertainties we find that the shapes of the distributions are different. More important, however, is the predominance of rather short scales in the data; On the basis of the fully developed cavities alone, £_Lmax would be estimated as 40-80 m which is after all small, recalling that the Debye length is ^ 0.2 m and a proton Larmor radius is ~ 2 m and ~ 7 m for oxygen ions. It is relevant to consider, for a comparison, the typical scale for lower hybrid cavities formed by the modulational instability by balancing the nonlinear term with the dispersive term (Shapiro et al., 1993), using (6). The resulting transverse length scale is An T /2 r R( * < M"£V nm i-imax "- R I —;~ , l2 f I , (loj where | V0 )£ is the square of the electric field in the center of the cavity at the initial stage. In case the high-frequency field is turbulent with a broad wavenumber range, we expect that the scale-length obtained here have to be shorter than the correlation length of the fluctuations. The B-parallel scale length becomes (17) With typical parameters from the data we find iimax ~- 104 m and in max "-' 101D In- Thus ^limax *s essentially infinite, i.e. much longer than the ?Uitnde of the satellite. Within this simple model for the temporal dynamics for the collap ,e, the value obtained for Limax will determine also the average distance between cavities in the B perpendicular direction, 1 c /'imax in Fifis 7 or 8 should be the same as the average distance in Fig. 3. Also for this scale length we observe a pronounced discrepancy between the theoretical value and the one obtained from the data. Evidently, the arguments in these sections are based on the assumption that the ciivily parameters L i and L\\ do not change appreciably during the time it takes the spacecraft to traverse a cavity. The density signals from the two probes confirm that this requirement is fully satisfied It is, however, not necessarily so for an arbitrary collapse scenario predicted from thr tnudrl equations (-1) and (5). A characteristic time-scale for the process (Shapiro et al . IWK1) ran hi* estimated by balancing the time derivative with the nonlinear term in \A) and iiMiig |f»), Kiving , m*l Ui0T, 'iff - *Ut ,; i «- • 3 (lM) 16 For our case this time scale is of the order of 1 ms, which is comparable to the time it takes the satellite to traverse a cavity at its maximum diameter. Within the analytical model bafed on eqs. (4)-(5), the assumption of quasi stationary cavities is thus marginally satisfied at best, and in disagreement with observations based on the two density probe signals. It is possible to L'patch-up" the disagreement with the scale-sizes of the collapse model by assuming that the electric field amplitude was considerably underestimated in the foregoing calculations. In this case, however, the analytical time-scales become much shorter, which is in pronounced disagreement with observations. In particular, the time record for the density depletions should then appear skew, or non-symmetric, if the cavities were rapidly contracting during the passage of the satellite. Significantly skew density variations are observed only very rarely. 4.4 Comment on wavelengths of lower-hybrid waves The most probable diameter of a cavity is of the order of 50 m according to Fig. 8. It can then be argued that in those cases where a trapped mode can be identified by the wavelet analysis, see Fig. 4, its local wavelength-component perpendicular to the magnetic field is most probably of the order of or less than ~100 m. Assuming that the free component propagates in the direction essentially perpendicular to B, we can conclude with reference to the dispersion relation, Fig. 2, that this component must have an even shorter wavelength, since its frequency is higher. The uncertainty in the basic plasma parameters entering the dispersion relation does not allow a reliable estimate of the actual wavelengths which may also be obscured by an unknown B-parallel component. 5 Discussions By comparing the observed probability densities with those derived on the basis of a simple collapse process we readily note a significant disagreement, which is most conspicuous both when characteristic times tm and scale sizes Limax are compared. As already mentioned, the two- point observations of density depletions indicate almost time stationary structures. Differences in measured cord-lengths from the two probes can fully be accounted for since (he two probes generally cross a given cavity at different positions with different cord- lengths- Cases where only one of the probes delect a cavity are similarly easy to explain by assuming that one is outside the cavity. In particular it is quite possible for thr electric field probes l<> detect a significant activity, while only onr of thr density probes indicates a cavity Kor ;i cylindrical cavity, the probability is 2 17 separation onto a direction perpendicular to the magnetic field and to the trajectory of the spacecraft. This result can actually also be used to obtain a crude estimate for the B- perpendicular scale size of the cavities having in mind that d = d(t) due to the satellite spin. The entire probability density for the cord length estimates with two probe measurements is also quite easy to obtain. The observed characteristic scales are difficult to accommodate within a simple collapse model. The characteristic time scale for a wave-collapse process is comparable to 1 ms which is much too short to be accounted for by the experimental results. The characteristic value for the maximum perpendicular length scale is calculated to be Zamax ^ ^4 m on the basis of the actual basic plasma parameters. This is much larger than the ~50 m obtained from the observation results in Fig. 8. Moreover this £±max is also larger than the average distance between cavities, an observation which is particularly difficult to fit into a model based on eqs. (4) and (5). We note that the experimentally obtained exponential distribution of cavity separations could well be maintained also within the model based on cavity formation as a result of a modulational instability, but requires in this case the cavities to have a significant spread in velocity. This is actually consistent with observations. Without such a distributed velocity, the distribution of cavity separations should be peaked around the average value. The average distance should, however, not be changed by a cavity propagation and the disagreement referred to before is significant. The estimate for i-j.niax 's based on the assumption that the process is initialized by a modulational instability. The data definitely seems to rule out this explanation. An alternative is that the wave excitation is burst-Like, starting with a large amplitude in a limited region of space. This explanation relies on the other hand on the experimental estimate for the wave amplitude to be significantly below the actual one. Also, it is emphasized that wave energy is always concentrated in density depletions al though the steady state relation (6) indicates that this need not he the case. This obser vation titotir is however not a strong argument against a collapse inudel, since smal' terms left out m the analysis in (4) and (5) will break this apparent symmetry. One possible model which retains some of the basic ideas of the collapse and still fits the data is one a«4iini)ns * very short collapsing phase which leaves small scale density depletions behind after burn out. These small cavities can then act as seeds for the next process which starts from these small scales. The process can continue repetitively and the observed cavities arc in this model the result of many subsequent collapses These must. however, stop «i a scale --40 80 m. The structures characterize . by this scale must then have a lunn lifetime to make their probabibty of detection dominant The model equations (4) anil l"»l are not able to account for this saturated stage and they do not contain any rhararterisiir length scale corresponding to the one observed Alternative ppn-e^se* for explaining the observations should be considered ttefernnn for instance I.. <>ur ray tracing cnlctilnttons, it cau be argued thai a» ihe wave vp,-niU a coin- 18 paratively long time in density cavities it is likely to give energy to the particles, ions as wel] as electrons although at different rates. Since the linear damping rate was found to be negligible, this must be assumed to be a nonlinear process. Effective particle heating (or rather energization) by the waves was demonstrated by for instance the numerical sim ulations of McBride et al. (1972), where a part of the ions are heated predominantly in B-perpendicular direction, the electrons in the parallel direction. The locally energized particles will leave the density depletion due to the locally enhanced pressure, the details of the process depending on the actual velocity distributions of the particles and the am- bipolar electric fields which are being set up. The local plasma density is consequently further depleted giving rise to stronger deceleration and trapping of waves, thus resulting in a positive feed-back which manifests itself as an instability. This argument assumes that particles are not energized strictly in the direction perpendicular to the magnetic field, but this is hardly a serious restriction. If the effect of slightly diverging magnetic field lines is taken into account actually even energization of particles in B-perpendicular direction can lead to a density depletion in a flux tube, due to the magnetic mirror force on the ions, as discussed by Singh (1994). It seems, however, that energetic particles are observed simul taneously with cavities only in some of the cases by the instruments on FREJA (André et al. 1994), in contrast to the first observations reported by Vago et al. 1992, where on the other hand also the observed density depletions were much deeper. This physical mecha nism may thus not be the only one operative. It might be emphasized that a small bulk ion or electron heating need not be detected at all by the instruments, even if the energization are sufficient for establishing the feed-back process described before. It is thus possible that the threshold for the energy analyzer on the TOPAZ III rocket was better than on FREJA (P. Kintner, private communication). The physical mechanism for cavity forma tion by the lower-hybrid waves outlined here is appealing because it indicates a preference for the well defined B-perpendicular length scale; the fastest way to deplete the density in a long "cigar" aligned with the magnetic field is by Allis diffusion, where the ions move perpendicular to the magnetic field while the electrons, with their small Larmor radius, move along B. Since the plasma is collision less to a good approximation, this process is only possible for B- perpendicular scales less than (or of the order of) twice the ion Larmor diameter of the heated ions. Of course also wider structures can be depleted, but in this case also the ions have to move along B, resulting in a much slower temporal development. Assuming that ions and electrons are healed by an equal amount, it is expected that a 1% density depletion is caused by a corresponding local 1% temperature increase. The wave energy density is usually smaller than 1% of the thermal energy density so the argument relies on wave energy flowing in at a continuous rate from outside the cavity. As wave activity is observed in the mf band also outside cavities, this requirement for consistency is fulfilled as well The mechanism mil lined in this section seems to deserve further scrutiny It is an example of a well known type of thermal instability which generally have low thresholds and small growth rates 19 5.1 Empty cavities When the lower-hybrid waves burn out, they leave an empty cavity. Such empty cavities are indeed observed, although one should bear in mind that it is seldom possible to verify that a density depletion is entirely void of lower-hybrid waves in the lf~ as well as the 7Ti/-band. The burn-out phase has to start gradually, and be a rather slow process, since according to the observations the density depletions reach a quasi-static stage and the only possibility for a further dynamic evolution of the cavities comes by damping of the lower-hybrid waves. For Te ^ T, we would expect that linear Landau damping would smear out the density depletions as soon as the maintaining lower hybrid wave has disappeared. The probability of observing empty cavities should consequently be small. It is not, and we conclude that the lifetime of the cavities is enhanced by other means. By numerical simulations in one spatial dimension we have evidence for the formation of BGK-type structures, phase space vortices, which from numerical simulations and laboratory experiments are known to have a long lifetime (Pécseli et al., 1984, Pécseli, 1984, Bostrom et al., 1989). The simulations assume fully kinetic ions with isothermally Boltzmann-distributed electrons in the electrostatic potential and a model for the ponderomotive potential. In a full three dimensional system with a magnetic field these structures can form long lived quasi-equilibria. Such nonlinear BGK-equilibria can for magnetized plp^mas be con structed trivially from the Vlasov equation in its appropriate form, i.e. the drift kinetic equation ft/ + rl-(vi/) + a/-^l*fii(/. • (i9) with vL V,* - 6 - n^'VJd,* + vtdt$)\fBa, where (19) is to be completed with Poisson's equation and equations for the electron dynamics. The second term in v± is the polarization drift which originates from the time variation of the E • B/#2-velocity experienced in the particle frame of reference. This effective acceleration, g, gives rise to a slow particle drift in the g • D-direction in or out of the potential well, resulting in a violation of the assumed tunc stationarity of the particle distribution In the Limit of large B-fields, i f. large i1rt, the polarization contribution to v. is negligible and a steady state solution to (19) is / F'^Mv: »• r A corresponding expression for «, /^ f*di', can be obtained similarly where the polar ization drift can be ignored from the outset In sonic problems it might safely be assumed that the electrons are isolhermally Boltzmann distributed at all tunes, which simplifies the relation bet ween n, and *fr If 'fr 'Mr,,.:) is prescribed then (20) ran be considered as nn equation fr.r determining /•' with standard methods available where F is assumed to 20 be composed of both free and of trapped particle contributions (Pécseli, 1993). The rx dependence of $ imposes then the r± variation of F. The solutions are not restricted to cylindrical symmetry. Little is known however about the stability of these solutions with respect to two or three dimensional perturbations. The arguments given before explicitly assumed large magnetic fields. It might be expected that they are applicable also for intermediate field strengths provided the polarization S term in v_t is small, i.e. VT,^X^*/^« * negligible, taking vrt) the ion thermal velocity, as representative for vz. In the extreme case with dt$ = 0 a cylindrical BGF-equilibrium can be obtained, although this limit is not particularly interesting. A BGK-equilibrium can not logically be recognized from the satellite data, in particular the B-parallel scale length is difficult to estimate. Indirect evidence can, however, be obtained by studying the measured ratio between the density and the electric field variation. If the B-parallel scale is finite, in the sense that it is shorter than the coUisiona) mean-free path, we expect that the electrons obtain an isothermal Boltamann-equilibrium and e$jTe ^ n/n0 for empty cavities. The validity of this relation can easily be tested in the data. A density depletion is thus expected to be electron-rich. For wave-filled cavities the situation is more complicated as this balance is influenced by the large amplitude wave. Then the relation between $ and n depends for instance on the absolute velocity of the cavity and the result is complicated even for the relatively simpler caseof Langmuir solitons (see e.g. Pécseli, 1985). It is possible to find evidence in the data for a finite B parallel length of the density depletions, by these arguments. Again, however, the statistical significance ': limited, in particular it is, as already mentioned, difficult to ensure that a cavity is void of a wave-components in the If- as well as m/"-bands. For the cast* where the scale length along B is essentially infinite, there is no unique relation between The formation time of a phase space vortex is essentially determined by the bounce time for ions moving along the magnetic field. It could thus be assumed that the predominance of small spatial scale» in the experimental data is explained by a short collapsing phase which saturates into h'ftg lived phase sr ace equilibria, which are no longer correctly described by a model equation Uke (5) This explanation assumes that the formation time of the quasi equilibria n so short that it escapes observations This is again unlikely tn be the rase since we expect the B parallel length scale» to be large, anil therefore the bounce time to br large 21 6 Conclusions In this paper we discussed the interpretation of localized bursts of lower-hybrid waves and correlated density depletions observed on the FREJA satellite (Dovner et al. 1994, Eriksson et al 1994). Particular attention was given to an explanation in terms of wave- collapse. We concluded that this interpretation, in its simplest form, can be ruled out, on the basis of a pronounced disagreement between the length and time scales predicted by the collapse-model and those observed in the data. The statistical arguments are based on three distinct elements. Two are purely geometric, where the cord-length distribution is determined for given cavity scales, together with the probability of encountering those scales. The third part of the argument is based on the actual time variation of scales predicted by the collapse model. The statistical assumption is basically that the cavities are uniformly distributed along the spacecraft trajectory and that they are encountered with equal probability at any time during the dynamic evolution. We believe that the cylindrical and ellipsoidal models discussed are sufficiently general to accommodate actual forms of a collapsing cavity. The time variation given by (8) is only an approximation at early times of the evolution of large cavities. This uncertainty can not be of importance as the large scales seems not to be significantly represented in the data anyhow. Since the measured electric field are somewhat uncertain, we have not discussed the statistics of this quantity in any detail. Various modifications of the simple coll apse-model were outlined. It was pointed out that it might be possible to retain the collapse as a basic element if small amplitude empty cavities which are remnants of earlier collapses serve as a local seed for a subsequent collapse of a lower hybrid wave burst. It seems however necessary to present arguments, as those in the foregoing section, for extended lifetimes of empty cavities to make this scenario convincing. The experimentally observed distribution of cord lengths in the cavities is actually ex plained best by assuming a large number of cavities, uniformly distributed in space, with a small spread in diameters around an average value of ~- 50 m. In that rase the information in Fig H is that the cavities start out with a B-pcrpendicular scale size very close to the one they end up with. A mechanism where the waves give up energy to ions as well as elec trons seems to be the best candidate for explaining this characteristic length scale, which is then a consequence of the thermal expansion of the plasma, with electron» streaming along B while ions expand across the magnetic field lines for transverse structures smaller than or comparable to twice the ion Larmor diameter of the healed ions. In the simples explanation of the observations it could be argued that the density cavities are generated by processes entirely independent of the lower hybrid waves and their modulation is solely caused by the phenomena accounted for by the ray tracing in eg Fig T> With this explanation we find it problematic to account for the apparent Mimlanty of the density depletions, i r ;i preference for a characteristic scale m their generation rrKvhaniMn must be argued With minor modifications, the statistical analysis presented in this work can be generalized also for studies of the possible evidence of Langmuir wave collapse in rocket or satellite data. In particular the ellipsoid approximation discussed here contains also the "pancake" model which is relevant for the Langmuir problem in weak magnetic fields. Acknowledgements This work was in part supported by the Norwegian National Science Foundation. Valuable comments by P. Shukla and R. Bingham are gratefully acknowledged. 23 References W.P Allis, S.J. Buchsbaum and A. Bees, Waves in Anisotropic Plasmas (M.I.T. Press, Cambridge, Massachusetts, 1963). M. André, P. Norqvist, A. Vaivads, L. Eliasson, O. Nordberg, A.I. Eriksson and B. Holback, Transverse ion energization and wave emissions observed by the FRE3 \ satellite. Geophys. Res. Lett. 21, 1915-1918 (1994). R. Bostrom, B. Holback, G. Holmgren and H. Koskinen, Solitary structures in the magne- tospheric plasma observed by VIKING. Phys. Scripta 39, 782-786 (1989). P.O. Dovner, A.I. Eriksson, R. Bostrom and B. Holback, FREJA multiprobe observations of electrostatic solitary structures. Geophys. Res. Lett. 21, 1827 1830 (1994). A.I. Eriksson, B. Holback, P.O. Dovner, R. Bostrom, G. Holmgren, M. André, L. Eliassen and P.M. Kintner, FREJA observations of correlated small-scale density depletions and enhanced lower hybrid waves. Geophys. Res. Lett. 21, 1843-1846 (1994) B. Holback, S.-E. Jansson, L. Ahlen, G. Holmgren, L. Lyngdal, S. Powel and A. Meyer, The Freja wave and plasma density experiment. Space Sei. Rev., JH94 (in press) G. Kaiser, A Friendly Guide to Wavelets (Birkhåuser, Boston, 1994). J. LaBelle, P.M. Kintner, A.W. Yau and B.A. Whalen, Large amplitude wave packets observed in the ionosphere in association with transverse ion acceleration J Geophys. Res. 91, 7113 7118 (1986). R. Lundin, G. Haerendel and S. Grahn, The FREJA project Geophys Hes Lett 21. 1823 1826 (199-1) J.P Lynov. P. Michelsen, H.L Pécseli and J J Rasmussen. Interaction between rlertron hoies in a strongly magnetized plasma. Phys Lett 80A. 23 25 (l!»8()| J. B. McBride, K. Ott, J. P. Boris and J H Orens, Theory and Mtiiulatmn "f turbulent heating by the modified two stream instability Phys Fluids 15. 23K7 23H3 U'»72) S.L Musher and B.I. Sturman. Collapse of plasma waves near the lower hybrid resonance Pij'nia Zh Kk*p Tcor Fw 22,537 5-12(1975) JKTP lett 22. 2»;5 *J«7( l'»75) L Nu'kenin and A Pirl. Direct observation of high frequency '-III^MO^ fnun n critical vrliK-ity rotating plasma Phy* Fluid* 30. 1H«*.% ]S*') | I'.IMT) 21 HL. Pécseli, Electron and ion phase-space vortices. In: Proceedings from the Second Symposium on Plasma Double Layers and Related Topics. Innsbruck, Austria, 5-6 July, 1984 ed. R. Schrittwieser and G. Eder, (Studia, Innsbruck, 1984) pp. 81-117. H.L. Pécseli, Solitons and weakly nonlinear waves in plasmas, IEEE Trans, on Plasma Sci. PS-13, 53-86 (1985). H. L. Pécseli, Coherent structures in plasma turbulence. In: Double Layers and Other Nonlinear Potential Structures in Plasmas. Proceedings of the Fourth Symposium on Double Layers. Innsbruck. Austria, July 6-8, 1992. Ed. R. Schrittwieser. (World Scientific, Singapore, 1993) pp. 104-113. H.L. Pécseli and J. Trulsen, A wavenumber-in-cell simulation of weak plasma turbulence. Phys. Scripta 46, 159-172 (1992). H.L. Pécseli, J. Trulsen and R.J. Armstrong, Formation of ion phase-space vortexes. Phys. 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Transverse ion acceleration by lower hybrid waves in the topside auroral ionosphere J Geophys Res. 07. 16935-16957 (1992). 25 Figure Captions Fig. 1. Example of lower hybrid wave cavity detected by instruments on the FREJA sateUite. The lower hybrid wave field is shown in a) and b) giving the low (//)- and medium {mf)- frequency bands of the detecting probe circuits. In c) the magnetic field variation is shown. In d) and e) the low frequency plasma density variations are shown as obtained by two Langmuir probes with an 11.0 m separation. The signal in a) with bandwidth 0-16 kHz and a sampling rate of 32 • 103 samples/s, the signal in b) with a bandwidth 0-2 kHz is obtained by a sampling rate of 4 • 103 samples/s. A frequency dependent phase change in filters does not allow a simple superposition of the signals. The spin-period of the satellite is 6 s, so its spin phase can be considered constant during a given data sequence. For the presentation here we used the output from the ISDAT program. Fig. 2. The lower hybrid wave dispersion relation for the parameters given in the text. The result is obtained by numerical solution of the full plasma kinetic dispersion relation with two 8 3 9 3 ion-species with densities nw+ = 3.8-10 m" and no* = i.210 m . The minimum value of kL is 0.01. CMose to the origin, the figure is dominated by the cold-plasma resonance cone. The transition to the magnetosonic mode is noticeable by a small "dip" at small wavenumbers. Fig. 3. Distribution of distance between cavities. Dotted line shows an exponential fit. A data gap appears for cavity separations smaller than a typical cavity width. Fig. 4. Wavelet analysis of the mj-signal of the waves inside and in the vicinity of a cavity with a) corresponding to the data in fig. 1. Figure b) shows another example with a trapped wave component indicated by an arrow. Note a small increase in frequency of the free mode inside the density depletion, which indicates a modified Doppler shift due to the variation in local wavenumber in the density depletion. In c) we show the // part of the signal in b). A Morelet wavelel was used in all cases. We find that if waves are detected in the If band then these will, with our interpretation, often correspond to a trapped component. Fig. 5. Typical result of a ray tracing analysis of lower hybrid waves propagating in a 1/2 % density depletion of Ctaussian shape with a width of 2000 m along the magnetic field (: axis) and ISO tn in the perpendicular direction. The initial value of the wavevectors are i, (I '>. k„ i> and k. 4 • 10 \ 3.5 • 10 4 and 3 10 \ respectively, at a position (J-.J/. .-) (200.11,»). 26 Fig. 6. Two examples for close, overlapping, cavities. These and similar observations can be interpreted as coalescing cavities. Fig. 7. Probability densities for cord lengths, t, in the case of cylindrical cavities in a) and ellip soidal ones in b). Fig. 8. Distribution of cord-lengths (not to be confused with cavity widths) obtained from the data. A gap in data for the smallest lengths appears because it is difficult to distinguish and identify very narrow density depletions, in particular when their amplitude is small. If such a depletion is represented by ~10 sampling points or less in the data it may not be possible to properly identify it. 27 F>ejg * -f-4- de pl 2 m-r Freja *• f4 dh/n pS |-f * 1 .OO - -W- A^ :_ o.so - L \ / s \ / : — o.so - \ / r -,.=o- r .0*t-00 *.0« — OS G.O« — OS 1.2« — O'i UCt^o2 2.0« — Ql J8?O.-i- i.=o- — 1 .SO - '•_ s - ~^"\ /"""~ -..==- ^^ r 1 1 1 ; • B"o—O0 4.O.-0! B^O.-OS^.i.-OZJJJ.-OZ 2.O.-0| • S?0.-^00 4.O.-0B 8.O.-03 1.2.-02 1.0»-OI *.o.-o Fr-eja "* -f-d- db ^ m-F * data gap OO d.O«-03 B.O« —03 1.4« —02 1.0«—02 2.0«- •6°o^r 9«eond9 frow 93D1D7 1 ©**-*«.O2 , df —O.OOO K » 39l0 3900 ^3890^ 5 3880 §3870^ *" 3860 3850 •%^-=>* ^s>**?-^ > Fig. 2 -i 1 r~ 0.25 h 0.2 0.15 r^ 0.1 \ 0.05- N a Efliririnr h - I 1=3 d. 0 100 200 300 400 500 600 700 800 DISTANCE (m) Fig. 3 FREJA Date: 07.01.1993 Time: 16h 44m 45s Wavelet Amplitude: Linear scale (art), units) 2 51 10.0- 9.0, 8.0 J 7.0 -| 6.0- 5.0 -j 4.0 -j i 3.0- 2.0- 1.0- 0.0 J 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 Time (ms) Wavelet-transform ofE FREJA Date: 07.01.1993 Time: 16h 57m 29s Wavelet Amplitude: Linear scale (art), units) 6.0 7.5 9.0 15.0 Time (ms) Wavelet-transform ofE FREJA Date: 07.01.1993 Time: I6h 57m 29s Wavelet Amphlude Linear scale (arb units) 20- 1.8- 16-, 1.4. 1.2 J 1.0 0.8-I 0.6- f 0.4 J 0.2 00 J 00 3.0 6.0 9.0 12.0 ISO 180 21.0 24.0 27.0 300 Tim» (ma) tt avtit 1-transform oflf-E Fin. t -2000 -1000 0 1000 3000 DISTANCE ALONG B (m) Fig. b Fraja - ~f*± dap 12 i -r ~ Z-.J3 -w— :*»^rtWfrv^^^^ — .°.-.^. Aswf %«.ra F *• s> ja " -*1 *A ar--' r*» oS i-^ FTQJQ - f4 dn/n p5 l-F •^ " \m\^\ \ * fi*Wt>W VV a* — o".*J>00 ^mfwi^r P .• e ; i " -^ -i d <• Ft'eja f4 dn/n p6 l-F rH J \" k .1 l :Oi J5 & • O.OO 0.O2 O.O^ Q.OS O.OS QSO' 0~ '6S - 3 - . ~* 1.4 1.2 • 1 0.8 O.é 0.4 • 0.2 lsm$/L±max l/L, Fig. 7 1 1 ,...._,_... 1 pi Pl n n n r - ,i._-..,.. 1 il - il Lii. i il i ' "•« u « « « il "w im i1 i » , n»'ii i nni ii n i i ni ii 1 0 20 40 60 80 100 120 WIDTH (m) * Fig. 8 FYSISK INSTITUTT DEPARTMENT OF FORSKNINGS- PHYSICS GRUPPER RESEARCH SECTIONS Biofysikk Biophysics Elektronikk Electronics Elementærpartikkelfysikk Experimental .Elementary Particle Physics Faste staffers fysikk Condensed Matter Physics Kjerne- og energifysikk Nuclear and Energy Physics Plasma- og romfysikk Plagma and Space' Physics Strukturfysikk Structural Physics Teoretisk fysikk Theoretical Physics ISSN 0333 S5*